1) Really dependant on the product/industry, but a bunch of numbers I've seen (and I've been looking into it a lot lately), have claimed from 10-40% trial to paid conversion increase by improving your onboarding emails. (Though the high end has come from also offering things like concierge onboarding to certain amenable customers.)
Improving your product's important, but you have keep in mind that the product is not the business, and a 10% improvement to your product likely won't have anywhere near the effect that a 10% improvement in your marketing will.
2) Sending a single welcome email with first steps that can be used as a reference, sending "trial ending" emails to remind people that their trial as running out (and offering an extension, usually), and 30-90 days after expiry emails offering a trial extension in case they got busy (happens very frequently) or asking for info why they didn't decide to go with your product. Anything above that, you want to start getting into tracking user lifecycle events and customizing your onboarding to that, and in fact, once you start doing that, you can start improving those low-hanging fruit emails as well.
3) You can use something as simple as Mailchimp to something as powerful as Salesforce. It really depends on what part of your funnel you want to focus on.
One thing people don't tend to get about email marketing before diving in is that there's no "one thing" that's email marketing. There's newsletter marketing, where you capture prospects, there's onboarding, there's "regular update" emails, there's winback emails, there's retention emails. Email's just a tool, really, and it's applicable and nearly all levels of marketing.
I've really gotten into this kind of stuff over the past few years, and if you still have questions, I'd be happy to chat with you about it. Email's in my profile.